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Sur la capitulation des 2-classes d’idéaux de Q(\sqrt{d} , i).

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Abstract

Let d ε N, i = √−1, k = Q (√d, i), k1 be the 2-Hilbert class field of k, k2 the 2-Hilbert class field of k1, Ck, 2 the 2-component of the class group of k and G2 the Galois group of k2/k. We suppose that the group Ck, 2 is of type (2, 2); then k1 contains three extensions Fi/k, i = 1, 2, 3. The aim of this Note is to study the problem of capitulation of the 2-ideal classes of k in Fi, i = 1, 2, 3, and to determine the structure of G2.

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... For a quadratic ield k, using Gauss's genus theory, one can easily deduce the rank of Cl 2 (k) . For biquadratic ields, the authors of [1,3] determined all positive integers d such that Cl 2 (k) is isomorphic to ℤ∕2ℤ × ℤ∕2ℤ , where k = ℚ( √ d, √ − ) and = 1, 2 . The papers [4,5] investigated the rank of Cl 2 (k) for the ields k = ℚ( √ d, √ m) , where m is a prime and d a positive square-free integer. ...
... The papers [4,5] investigated the rank of Cl 2 (k) for the ields k = ℚ( √ d, √ m) , where m is a prime and d a positive square-free integer. In the same direction, [7] classiied all ields k = ℚ( √ d, i) such that Cl 2 (k) is isomorphic to ℤ∕2ℤ × ℤ∕4ℤ or (ℤ∕2ℤ) 3 . In [9,11], E. Brown and C. Parry determined some imaginary quartic cyclic number ields k such that the rank of Cl 2 (k) is at most 3. Finally, [24] determined all imaginary biquadratic ields whose 2-class group is cyclic. ...
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